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Bayes' Rule

Author: Sophia

what's covered
This tutorial will cover the topic of Bayes' Rule. Our discussion breaks down as follows:

Table of Contents

1. Bayes' Rule

Bayes' Rule is a theorem that allows us to turn around a conditional probability statement. This rule allows you to update or revise your probabilities in light of new information. Essentially, it reverses the conditional probability formula, allowing you to find the probability of A given B from the probability of B given A.

formula to know
Bayes' Rule
P left parenthesis A space vertical line space B right parenthesis equals fraction numerator P left parenthesis A right parenthesis space P left parenthesis B space vertical line space A right parenthesis over denominator P left parenthesis B right parenthesis end fraction

term to know
Bayes' Rule
A reversal of the conditional probability formula that asks "given that this second event happened, what is the probability that this other event occurred first?"

1a. Game Show Example

Suppose there's a famous television show called "Go For Broke.” On the show, a person gets to roll a die to determine which jar they're going to draw a colored chip from. If the chip is green, they will win a prize, and if the chip is red, they won’t win anything.

  • If the contestant rolls a 1 or a 2, they select from jar A, which contains nine green chips and 12 red chips.
  • If they roll a 3, 4, 5, or 6, they have to select from jar B, which only contains five green chips and 25 red.
File:9679-JarA.png File:9668-Jar_B.png
Roll a 1 or 2:
Pick from Jar A
Roll a 3, 4, 5, or 6:
Pick from Jar B

Rolling a 1 or a 2 and being able to select from jar A gives the contestant a better probability of winning a prize. Here is the tree diagram of the probabilities in this game:

Tree Diagram: Dice and Chips

  • Contestants have a 1/3 probability of rolling 1 or 2 and selecting from Jar A.
  • If they are selecting from jar A, they have a 9/21 probability of getting the green chip and a 12/21 probability of getting the red chip.
  • Contestants have a 2/3 probability of rolling 3, 4, 5, or 6 and selecting from Jar B.
  • If they are selecting from jar B, they have a 5/30 probability of getting the green chip and a 25/30 probability of getting the red chip.
By multiplying out the values on the tree diagram, we find that we get these values for the probability of choosing either color chip from either jar:

Tree Diagram: Dice and Chips Percentages

think about it
Suppose you tuned in late to Go for Broke one day and didn't get to see which jar the chip came from. All you see is that the contestant won the prize by getting the green chip.

Given they got a green chip, what's the probability that they rolled the 1 or 2, meaning they selected their winning chip from jar A?

To answer that question, what you're actually trying to find is the probability that you would have picked from A, given that you got the green chip. This requires the following conditional probability statement:

P left parenthesis A space vertical line thin space G right parenthesis equals fraction numerator P left parenthesis A space a n d space G right parenthesis over denominator P left parenthesis G right parenthesis end fraction

But recall that the "and" probability of two dependent events is the probability of one event times the conditional probability of the second event, given the first event occurred. So we can actually rewrite this equation:

P left parenthesis A space vertical line space G right parenthesis equals fraction numerator P left parenthesis A space a n d space G right parenthesis over denominator P left parenthesis G right parenthesis end fraction equals fraction numerator P left parenthesis A right parenthesis times P left parenthesis G space vertical line thin space A right parenthesis over denominator P left parenthesis G right parenthesis end fraction

When rewritten in this new formula, you may notice that the probability of G given A helps us to find the probability of A given G. When we actually go through and run the scenarios, the probability of A and G was one of the branches on the tree diagram, or 1/7.

To find the probability of G, we can see from the tree diagram that this can happen in two ways: selecting from jar A or jar B.

Plugging in this information, the equation simplifies down to the fraction 9/16::

P left parenthesis A space vertical line space G right parenthesis equals fraction numerator P left parenthesis A right parenthesis times P left parenthesis G vertical line A right parenthesis over denominator P left parenthesis G right parenthesis end fraction equals fraction numerator begin display style 1 over 7 end style over denominator begin display style 1 over 7 end style plus begin display style 1 over 9 end style end fraction equals fraction numerator begin display style 1 over 7 end style over denominator 16 divided by 63 end fraction equals fraction numerator 9 divided by 63 over denominator 16 divided by 63 end fraction equals 9 over 16

What this means is that out of every 16 times you pick the green, 9 of those times it came from jar A. This is a surprising conclusion! The probability that you would pick from jar A, given that you ended up winning, is actually over half.

1b. Finding a Crashed Plane

Another example of when you might use Bayes' theorem is during an actual army practice to find the wreckage of a plane.

Suppose that a plane was going from Miami to Mexico City, but then it crashed somewhere in the Gulf of Mexico. Here is a map of the scenario:

Map of Crashed Plane

As the map shows, the plane may have lost radio contact in one location. Therefore, because they don't know what else might have happened with the plane, the search party might start searching in that area. However, perhaps a few days go by, and people find debris in a different location.

Bayes' theorem allows you to analyze the probability that the plane crashed in the region where they lost radio contact, given that the debris was found elsewhere, where the green circle is.

If that probability is low, you might choose to start searching in the green region, assuming that the plane tried to go around the storm by going north.

The idea is that we can update our guesses based on new information.

summary
Bayes' rule is a mathematical theorem that allows you to amend probability statements based on new information. With this rule, you can turn around the conditional probability statement and find the probability of A given B from the probability of B given A. Therefore, it lets you use newer knowledge to adjust probabilities of prior events.

Good luck!

Source: THIS TUTORIAL WAS AUTHORED BY JONATHAN OSTERS FOR SOPHIA LEARNING. PLEASE SEE OUR TERMS OF USE.

Terms to Know
Bayes' Rule

A reversal of the conditional probability formula that asks "given that this second event happened, what is the probability that this other event occurred first?"